## Abstract

We prove the undecidability of mso on ω-words extended with the second-order predicate U_{1}(X) which says that the distance between consecutive positions in a set X ⊆ N is unbounded. This is achieved by showing that adding U_{1} to mso gives a logic with the same expressive power as mso+U, a logic on ω-words with undecidable satisfiability. As a corollary, we prove that mso on ω-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e., sets X such that for some positive integer p, ultimately either both or none of positions x and x + p belong to X.

Original language | English |
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Article number | 12 |

Journal | Logical Methods in Computer Science |

Volume | 16 |

Issue number | 1 |

DOIs | |

Publication status | Published - 11 Feb 2020 |

## Keywords

- MSO logic
- Undecidability